Rigorous Model for the Simulation of Gas Adsorption and Its

ARTICLE pubs.acs.org/IECR

Rigorous Model for the Simulation of Gas Adsorption and Its Verification W. B. Li, B. T. Liu,* K. T. Yu, and X. G. Yuan State Key Laboratory for Chemical Engineering and School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China ABSTRACT: A rigorous model containing conservation equations of momentum, energy, and mass, with their kε, t2εt, and newly developed c2εc closing formulations was proposed for simulating the gassolid adsorption process. The feature of the proposed model is that the unknown turbulent mass-transfer diffusivity Dt need not be estimated in advance. With this model, the breakthrough, regeneration, and temperature curves, as well as the Dt, were simulated for the methylene chloride adsorption and regeneration processes. The simulated results were closely checked by the published data. It was found that the Dt was unevenly distributed in the axial and radial directions, which means that the Schmidt number or Peclet number is varying throughout the adsorption column and cannot be simply represented by a constant or a simple formula. The simulated results on axial and radial concentration distributions along the column at different time are helpful to understand the process dynamics. The proposed model is especially useful for simulating the adsorption process where the turbulent mass-transfer diffusivity is unavailable.

’ INTRODUCTION Adsorption has long been known as a common technology covering a wide range of different separation and purification processes in the chemical and related industries, such as the separation of hydrocarbon mixtures,1 the desulfurization process for natural gas,2 and the removal of mercury emissions from coal combustion flue gas.3 Considerable research has been focused on the experimental measurement of a breakthrough curve for studying the dynamics and effectiveness of the adsorption process. The modeling of the adsorption process is complicated, as the flow and concentration distribution of the adsorbate is nonideal. Most of the research has been based on a one-dimensional413 or two-dimensional dispersion model,14,15 in which the adsorbate flow is either simplified or computed by using computational fluid dynamics (CFD),16 and the effect of uneven adsorbate concentration is represented by adding axial and radial dispersion terms with undetermined turbulent mass-transfer diffusivities Dt,a and Dt,r. The usual way to find the unknown diffusivity is either by employing empirical correlation obtained from inert tracer experiment or by guessing a Schmidt number applied to the whole process. However, from a theoretical viewpoint, the diffusivity obtained by using an inert tracer technique, which involves no adsorption, is different from that with adsorption. On the other hand, the guess of a constant Schmidt number on the basis of experience is unreliable. Thus, the evaluation of the unknown turbulent masstransfer diffusivity on a sound base for accurate modeling of adsorption process is still lacking. One way to obtain the diffusivity directly is to close the masstransfer differential equation by a proper method in order to solve at once all unknown parameters in the equation including the turbulent mass-transfer diffusivity. Recently, a c2εc model17 was developed for this purpose by adding two accompanied equations to the mass-transfer differential equation, namely, the concentration variance c2 equation and its dissipation εc equation. r 2011 American Chemical Society

This model has been applied successfully to predict the turbulent mass-transfer diffusivity and Murphree tray efficiency of a commercial-scale distillation column.18,19 Liu et al.20,21 also employed this model for simulating the absorption of CO2 by NaOH and monoethanolamine (MEA) in a packed column with success including the prediction of turbulent mass-transfer diffusivity. The fact that all of the above simulated results have been verified by experimental work indicates that the c2εc model is applicable to the mass-transfer processes without knowing in advance the turbulent mass-transfer diffusivity. Therefore, an attempt was made in this study for extending its application to the adsorption process. The present rigorous model consists of the differential masstransfer equation, closing with c2εc equations, and the accompanied formulations of CFD and computational heat transfer (CHT). In the mathematical formulation of the accompanied CFD and CHT, the conventional methods of kε and t2εt are used for closing the momentum and heat-transfer equations. The present work includes the simulation of breakthrough and regeneration curves, the velocity and concentration distributions, the prediction of the local turbulent mass-transfer diffusivity, and the local Schmidt and Peclet numbers along the packed column as well as the temperature curves at different packed height. The simulated results are testified to by comparison with the published experimental data reported by Hwang et al.22 for the methylene chloride adsorption and regeneration processes.

’ ASSUMPTIONS The assumptions made for simulating the gas adsorption process in a packed column are as follows: (i) The gas phase is incompressible. Received: February 26, 2011 Accepted: May 26, 2011 Revised: May 3, 2011 Published: May 26, 2011 8361

dx.doi.org/10.1021/ie200360d | Ind. Eng. Chem. Res. 2011, 50, 8361–8370

Industrial & Engineering Chemistry Research

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(ii) The gas-phase mass flow is constant. This assumption is reasonable because the concentration of adsorbate is very low. (iii) The physical properties of both phases are constant. (iv) The adsorbent particles are of spherical form. (v) The mass-transfer, heat-transfer, and flow behaviors are axially symmetrical in a packed column; that means the proposed model is symmetrical two-dimensional. (vi) Because the column wall is very thin, the temperature is uniform along the radial direction in the wall. In addition, as Tobis and Ziolkowski23 have stated, the transition from laminar to turbulent flow occurs at Rep = 100. In the present study, Rep is about 130, which means our simulation is under turbulent flow condition. Model Equations. The proposed model consists of three sets of differential equations, expressed in tensor form, as follows: (1) Mass Balance Equation Set. differential mass-transfer equation for the adsorbate: " # D DC DðFγCÞ DðFγCui Þ þ ¼ FγðD þ Dt Þ þ Sc Dxi Dxi Dxi Dt

ð1Þ

equation of concentration variance c2: " # DðFγc2 Þ þ DðFγc2 ui Þ ¼ D Fγ Dt þ D Dðc2 Þ Dxi Dxi Dxi σc Dt DC DC 2Fγεc Dxi Dxi

kc2 Dt ¼ Cc0 k εεc

! þ Sm

where μ represents the molecular viscosity of the gas phase, Sm is the source term for the gas flow, and ui0uj0 is the Reynolds stress, for which the Boussinisque’s relation is applied: ! Du Du 2 j i δij Fk þ ð7Þ Fui 0 uj 0 ¼ μt 3 Dxj Dxi μt ¼ Cμ F

k2 ε

ð8Þ

where μt is the turbulent viscosity of the gas phase. To solve eq 8, the standard kε model is used as follows:

DðFγkÞ DðFγkui Þ þ Dt Dxi ! D μt Dk Dui Duj Dui ¼ γ μþ þ Fγε þ γμt Dxi σ k Dxi Dxj Dxi Dxj

ð9Þ

! D μt Dε ε Dui Duj Dui ε2 þ C1ε γ μt ¼ γ μþ þ C2ε Fγ ð10Þ σε Dxi Dxj Dxi Dxi k Dxi k

ð2Þ

where the constants are as follows:24 Cμ = 0.09, C1ε = 1.44, C2ε = 1.92, σk = 1.0, and σε = 1.3. (3) Energy Balance Equation Set. energy balance equation for the gas phase: DðFγcp, g Tg Þ DðFγcp, g ui Tg Þ DTg D þ ¼ Fγcp, g Reff þ STg ð11Þ Dt Dxi Dxi Dxi

ð3Þ

!1=2 ð4Þ

where Tg is the average temperature of the gas phase; cp,g is the specific heat of the gas phase, STg is the thermal source term; Reff is the effective thermal diffusivity of the gas phase, defined as Reff = R þ Rt, in which R and Rt are the molecular and turbulent thermal diffusivities, respectively; and Rt can be calculated by using the t2εt closing equation proposed by Nagano25 and simplified by Liu et al.20,21 as follows:

the constants in eqs 2, 3, and 4 are as follows:20,21 Cc0 = 0.11, Cc1 = 2.2, Cc2 = 2.2, Cc3 = 0.8, σc = 1.0, and σεc = 1.0. (2) Fluid Dynamics Equation Set.

equation of temperature variance t2:

conservation equation:

" 2 # DTg DTg D Rt Dðt Þ ¼ Fγ þR 2Fγεt þ 2FγRt Dxi Dxi σt Dxi Dxi

DðFγÞ DðFγui Þ þ ¼0 Dt Dxi

ð6Þ

DðFγεÞ DðFγεui Þ þ Dt Dxi

! " # DðFγεc Þ DðFγεc ui Þ D Dt Dðεc Þ þ ¼ Fγ þD Dt Dxi Dxi Dxi σ εc εc DC DC εc 2 εεc Cc2 Fγ Cc3 Fγ 2 2 k c Dxi Dxi c equation for computing Dt:

Dp D μ DðFui Þ ¼ γ þ γ Fγui 0 uj 0 Dxi Dxj F Dxj

equation of turbulent kinetic energy dissipation ε:

equation of concentration variance dissipation εc:

þ Cc1 FγDt

DðFγui Þ DðFγui uj Þ þ Dt Dxj

equation of turbulent kinetic energy k:

where C is the average mass fraction of adsorbate, F is the gasphase (adsorbate-phase) density, γ is the porosity of the packed column, D and Dt are the molecular and turbulent mass-transfer diffusivities, respectively, and Sc is the source term of mass transfer of the adsorbate. Dt is determined by the c2εc model,20,21 which consists of the following equations:

þ 2FγDt

momentum transport equation:

ð5Þ

DðFγt 2 Þ þ DðFγt 2 ui Þ Dxi Dt

8362

ð12Þ

dx.doi.org/10.1021/ie200360d |Ind. Eng. Chem. Res. 2011, 50, 8361–8370


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equation of dissipation rate εt: ! " # DðFγεt Þ DðFγεt ui Þ D Rt Dðεt Þ þ ¼ Fγ þR Dt Dxi Dxi Dxi σ εt þ Ct1 FγRt

be calculated by26 1 1 1 ¼ þ KG kf γp kp

εt DTg DTg εt 2 εεt ð13Þ Ct2 Fγ Ct3 Fγ k t 2 Dxi Dxi t2

equation for computing Rt: kt 2 Rt ¼ Ct0 k εεt

!1=2 ð14Þ

The model constants in t2εt equations are as follows:25 Ct0 = 0.11, Ct1 = 1.8, Ct2 = 2.2, Ct3 = 0.8, σt = 1.0, and σεt = 1.0. energy balance equation for solid phase: D½Fs ð1 γÞcp, s Ts D DTs ¼ F ð1 γÞcp, s ks þ ST s Dxi s Dxi Dt

ð15Þ

where Fs is the particle density, ks is the thermal conductivity of the solid phase, Cp,s is the specific heat of the solid phase, and STs is the thermal source term. The concentration and the temperature as well as the velocity distributions in the absorption column can be obtained by the simultaneous solution of the foregoing model equation sets. The evaluation of source terms Sc, Sm, STg , and STs and relevant physical properties F, γ, D, and μ in the equation sets are shown in the following sections. Evaluation of Source Terms. Since the model equations are solved numerically, the model equations are first discrete into a large number of small finite elements. Thus, the source terms are referred to the conditions appearing in the finite elements. From this point of view, the source terms can be formulated by using proper correlations under the individual conditions of the element, such as velocity, concentration, and temperature obtained in the course of numerical computation. (1) Source Term Sc of the Gas Phase. Source term Sc in eq 1 is given by ð16Þ Sc ¼ Fg KG ap MðY Y Þ where Fg is the total gas concentration, ap is the surface area per unit volume of packed column, M is the molecular weight of the adsorbate, Y is the mole fraction of the adsorbate in the gas phase, and Y* is the mole fraction of the adsorbate in equilibrium with the solid phase, which can be calculated by the following correlation for the gassolid equilibrium isotherm:22 q , q < qs ðTg Þ Y ¼ ð17Þ ðqs ðTg Þ qÞbðTg ÞP where qs is the saturation mole concentration of adsorbate, b is the Langmuir isotherm constant, P is the total pressure of the gas phase in the column, and q is the mole concentration of adsorbate in the solid phase, which can be calculated by Z Fg KG ap ðY Y Þ dt q ¼ q0 þ ð18Þ Fs

ð19Þ

where γp is the particle porosity. The intraparticle mass-transfer coefficient kp can be calculated according to a lumped model proposed by Gluekauf27 5Dp kp ¼ ð20Þ Rp where pore diffusivity Dp is given by Yang28 and Rp is the packing radius. The external mass-transfer coefficients kf were estimated with the correlation of Wakao and Funazkri:29 k f dp ¼ 2:0 þ 1:1Rep 0:6 Sc0:33 ð21Þ D where Rep is the Reynolds number based on the packing diameter. (2) Source Term Sm. Source term Sm in eq 6 represents the gravity and the resistance of gas flow due to the solid particles in a fixed column which is given as follows:30 Sm ¼ γðFGS þ FgÞ

FGS

! 150 μ ð1 γÞ2 1:75F ð1 γÞ ¼ þ juj u dp γ3 γ3 dp 2

ð22Þ

where column porosity γ can be calculated by using the following expression:31 γ ¼ γ¥ þ

" ! # 1 γ¥ 2π Rr Er ð1 0:3Pd Þ cos þ 0:3P d 2 aγ þ 1:6Er 2 Pd dp

ð23Þ where γ¥ is the porosity in an unbounded packing, R is the radius of the column, r is the position in a radial direction, and Er is the exponential decaying function, which is given by 2 !3=4 3 R r 5 Er ¼ exp41:2Pd ð24Þ dp where Pd is the period of oscillation normalized by the nominal particle size and Pd = 0.94 for sphere particle; aγ is a constant depending on the ratio of the particle size to the column size: 2 !3=4 3 2R R 5 aγ ¼ ð25Þ 1:6 exp42:4Pd nγ Pd dp dp 9 8 > > > > > > > > > > > > < 2 R = 2 3 nγ ¼ int !3=4 P d > d p> > > R > > > > 42:4Pd 5 1 þ 1:6 exp > > > > ; : dp

ð26Þ

(3) Source Terms STg and STs . Source term STg in eq 11 represents the amount of heat transferred from solid adsorbent to the gas phase:

where q0 is the initial mole concentration of adsorbate in the solid phase and KG is the overall mass-transfer coefficient, which can

STg ¼ hap ðTs Tg Þ 8363

ð27Þ



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Source term STs in eq 15 includes the heat produced by adsorption and the heat transferred to the gas phase: STs ¼ ΔHFs ð1 γÞ

Dq hap ðTs Tg Þ Dt

Table 1. Propertiesa of the Adsorption Column and the Adsorbent Particles property

ð28Þ

where ΔH is the heat of adsorption and h is the heat-transfer coefficient from solid adsorbent to the gas phase, which can be calculated by kg Nu h¼ Rp

ð29Þ

R (m)

0.41

cp,g (J (kg K)1)

970

Z (m) dp (m)

0.20 0.002

cp,s (J (kg K)1) T0 (K)

836 298

Fs (kg m3)

420

ks (W (m K)1)

0.3

γp

0.67

ΔH (J mol1)

28020

γ¥

0.42

a

where Nu is the Nusselt number which can be estimated by the following correlation:32 Nu ¼

property

0:357 Rep 0:641 Pr 1=3 2γ

ð30Þ

R = inside diameter; Z = packed column height; dp = average particle size; Fs = bulk density; γp = particle porosity; γ¥ = average column porosity; cp,g = specific heat of gas; cp,s = specific heat of adsorbent; T0 = ambient temperature; ks = thermal conductivity of adsorbent; ΔH = heat of adsorption of adsorbate.

Evaluation of Relevant Physical Properties. In this paper, the molecular diffusivity D was calculated from the Chapman Enskog formula.33 Since the concentration of adsorbate in the gas phase is very low, the gas mixture viscosity μ, thermal conductivity kg, and density F are approximately equal to the those of pure carrier gas, which is air for the adsorption step and nitrogen for the desorption step. The viscosity and thermal conductivity of the pure gas are calculated by the method of Lemmon and Jacobsen,34 and the density F is obtained by the method of Lee and Kesler, which was reported by Perry and Green.35 Boundary Conditions. (1) Inlet Conditions. The inlet boundary conditions for the three sets of model equations are as follows:21 2 C ¼ Cin , kin ¼ 0:003uin , kin 1:5 2 εin ¼ 0:09 , cin ¼ ð0:082Cin Þ2 , dH εin cin 2 , Tg ¼ Tc, in , εc, in ¼ Rτ kin εin 2 ¼ ð0:082ΔTÞ , εt, in ¼ Rτ tin 2 kin

u ¼ uin ,

tin 2

Figure 1. Comparison of simulated breakthrough curve with experimental measurements (Hwang et al. data from ref 22): Yin = 2.25 103, Tin = 298 K, P = 1.10 atm, and F = 33.5 L min1.

ð31Þ

where dH is the hydraulic diameter of random packing, which can be calculated by36 dH ¼

4γ¥ ap ð1 γ¥ Þ

ð32Þ

The surface area per unit volume of the packed column ap can be calculated by30 ap ¼

6ð1 γ¥ Þ dp

ð33Þ

inlet for the adsorption process is usually small. However, for the desorption process the gas phase at the inlet is increased greatly with time, so ΔT is set to be 10 K for the convenience of computation. (2) Outlet Conditions. The outlet of flow is considered to be close to fully developed so that zero normal gradients are chosen for all variables except pressure. (3) Wall Conditions. The no-slip condition of flow is applied to the wall. However, the zero flux condition cannot satisfy the present simulation. Consequently, the more precise wall conditions which proved to be appropriate in this work are adopted as follows: 2 cw 2 ¼ ð0:082Cin Þ ,

37

and the term Rτ is the time scale ratio, which can be calculated by !2 kin 2 1 Rτ ¼ C t ð34Þ εin Dt 3

tw 2 ¼ ð0:082ΔTÞ2 ,

εt, w

where Rτ can be calculated by

2 1

In the present study, Dt was assumed to be 10 m s by trial, the correctness of which can be checked by the simulated result, and the value of Rτ may be recalculated again with the new Dt if necessary. For the selection of ΔT at the inlet condition, 0.1 K was chosen because the temperature change of gas phase at the

Rt, w ¼

εw 2 cw , kw εw ¼ Rτ, w tw 2 kw

εc, w ¼ Rτ, w

kw 2 1 Ct ε w Dt

ð35Þ

!2

kw and εw can be obtained by the standard wall function of the kε model. The near-wall urbulent kinetic energy kw is calculated by the 8364



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Figure 2. Sequences of contours of mole fraction at different time (where, for example, 2.25e-03 represents 2.25 103): Yin = 2.25 103, Tin = 298 K, P = 1.10 atm, and F = 33.5 L min1.

Figure 3. Contour of turbulent mass transfer diffusivity and turbulent Schmidt number at t = 75 min (where, for example, 9.13e-03 represents 9.13 103): Yin = 2.25 103, Tin = 298 K, P = 1.10 atm, and F = 33.5 L min1.

Figure 5. Profiles of turbulent Schmidt number at different packedcolumn heights at t = 75 min: Yin = 2.25 103, Tin = 298 K, P = 1.10 atm, and F = 33.5 L min1.

Figure 4. Profiles of turbulent mass-transfer diffusivity at different packed-column heights at t = 75 min: Yin = 2.25 103, Tin = 298 K, P = 1.10 atm, and F = 33.5 L min1.

Figure 6. Profiles of turbulent viscosity at different packed-column heights at t = 75 min: Yin = 2.25 103, Tin = 298 K, P = 1.10 atm, and F = 33.5 L min1. 8365



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Figure 7. Profiles of turbulent Peclet number at different packedcolumn heights at t = 75 min: Yin = 2.25 103, Tin = 298 K, P = 1.10 atm, and F = 33.5 L min1.

Figure 9. Simulated and experimental regeneration curves (Hwang et al. data from ref 22): Yin,ads = 2.50 103, Tg,in = 365[365 exp(0.352t0.666 1.654)] K, P =1.09 atm, and F = 33.5 L min1.

are the temperatures of the inner and outer walls. Considering the thin thickness and high solid thermal conductivity of the column wall, it can be assumed that Tw1 = Tw2 and Qw1 = Qw2 = Qw. Then we have Qw ¼ hw ðTg T0 Þ hw ¼

hw1 hw2 hw1 þ hw2

ð39Þ ð40Þ

where hw1 can be calculated by39 hw1 ¼ 0:023

kg ðRecol Þ0:8 ðPrÞ0:3 dcol

ð41Þ

and hw2 by40 hw2 ¼ bh Figure 8. Radial profile of interstitial velocity of the gas phase at z = 0.10 m: Yin = 2.25 103, Tin = 298 K, P = 1.10 atm, and F = 33.5 L min1.

k equation of the standard kε model, and the εw can be calculated by38 εw ¼

Cμ 3=4 kw 3=2 kyw

ð36Þ

where k, the Karman constant, is 0.4187; yw denotes the distance from the center of the near-wall grid to the wall. The heat loss of the adsorption column wall to the environment should be considered. The heat flux from the gas phase to the inner wall and the heat flux from the outer wall to the environment can be calculated by Q w1 ¼ hw1 ðTg Tw1 Þ

ð37Þ

Q w2 ¼ hw2 ðTw2 T0 Þ

ð38Þ

where hw1 and hw2 represent the heat-transfer coefficient from the gas phase to the wall and the heat coefficient from the wall to the ambient surroundings, respectively, Tg is the gas-phase temperature, T0 is the ambient temperature, and Tw1 and Tw2

kg ðGr PrÞn Z

ð42Þ

where kg is the thermal conductivity of the gas phase, dcol is the packed column inner diameter, Recol is the Reynolds number based on the packed-column diameter, Pr is the Prandtl number, Z is the height of the column, Gr is the Grashof number, and bh and n are heat convection parameters. The zero flux condition is applied to the mass-transfer equation. Simulated Results and Verification of the Proposed Model. The simulation was performed with the commercial software Fluent 6.3.26 (2D). The model equations were solved numerically with finite volume method. The SIMPLEC algorithm is employed for the discretization of momentum equation with the kε model. Considering the flow is symmetrical, only half of the packed column was simulated. Adsorption of Methylene Chloride. The object of the present simulation was the methylene chloride vapor adsorption on an activated carbon column, and the simulated results were compared with published experimental data.22 The adsorption column and adsorbent particle properties are given in Table 1. In the present simulation, the grid arrangement of the adsorption column comprised a total of 32 000 quadrilateral cells. In the radial direction, the nonuniformly distributed mesh was applied with higher grid resolution in the near-wall region. Such grid resolution provided sufficient simulation precision. 8366



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Figure 10. Sequences of concentration contours in mole fraction at different time (where, for example, 1.14e-02, represents 1.14 102): Yin,ads = 2.50 103, Tg,in = 365[365 exp(0.352t0.666 1.654)] K, P = 1.09 atm, and F = 33.5 L min1.

(1) Breakthrough Curve. The comparison between the simulated breakthrough curve and the experimental data is shown in Figure 1. The simulated curve by using the present model demonstrates not only matching close to the experimental result but also avoiding the use of any means to estimate the Dt. (2) Concentration Profiles along the Column. The contours of methylene chloride in mole fraction at different times are given in Figure 2, from which we can see how the concentration profiles in the column are developed and consequently affect the breakthrough curve. For instance, Yout/Yin is almost equal to zero at 15 and 45 min, while large amounts of adsorption have been undertaken in the column. It should be noted that the methylene chloride concentration distributions along axial direction are uneven, and the contours of parabolic shape were found in the radial direction. The nonuniform concentration distribution of adsorbate throughout the column is due to the existence of flow dispersion, uneven porosity, and the wall effect, which has been considered and modeled in the present simulation. Besides, Figure 2 also shows the details of the progress of the adsorption process at different times. The advance of adsorption in the column from 15 to 45 min is much faster than from 105 to 135 min. This result is helpful if the adsorption process should be optimized. (3) Profiles of Mass-Transfer Diffusivity. Figure 3 shows the profile of turbulent mass-transfer diffusivity Dt and turbulent Schmidt number Sct in the packed column at t = 75 min. Figure 3a shows the variations of Dt along radial and axial directions, which inspire us that the mass-transfer behavior in an adsorption column is complicated and cannot be much simplified as is usually done. For instance, Sct is usually assumed to be 0.7,37 yet, in the work of Frigerio et al.,41 the assumption of an Sct of 0.4 gave the best agreement with the experiment results. In the present work, the Sct calculated varied between 0.025 and 0.345 in the bulk region. Figure 4 gives profiles of Dt at five heights of the column, from which the wall effect was obviously observed. Winterberg et al.11 verified the existence of wall effect and proposed an approach including a set of equations to determine the radial distribution of Dt and showed that it was kept constant in the bulk and decreased in the near-wall region. Figure 4 shows even more complicated distribution of Dt, which is varying in both radial and axial directions, especially increases very sharply in the close vicinity of the wall. Such a result is different from Winterberg’s work; it may be due to the fact that in the present model Dt is influenced by both

Figure 11. Simulated and experimental temperature curves at different packed-column height (Hwang et al. data from ref 22): Yin,ads = 2.25 103, Tg,in = 365[365 exp(0.352t0.666 1.654)] K, P = 1.09 atm, and F = 33.5 L min1.

velocity and concentration fluctuations, which is more rigorous than that if the effect of concentration fluctuation is ignored. Figure 4 also shows that Dt is gradually increased when approaching to the top of the column, which is probably due to the rising of concentration fluctuation with increasing packed-column height. The distribution of Sct along the axial direction (Figure 5) demonstrated that the closer to the wall, the smaller Sct was. This could be explained by the increase of the turbulent mass diffusivity (Figure 4) and the decrease of the turbulent viscosity (Figure 6) near the wall of the column. The distribution of the turbulent Peclet number Pet along the axial direction is shown in Figure 7. Since D was much smaller than Dt, the axial mass-transfer diffusivity Da is approximately equal to Dt because Da = D þ Dt ∼ Dt. Thus, the turbulent Peclet number (Pet = udp/Dt) is also approximately equal to the axial Peclet number (Pea = udp/Da). The values of the axial Peclet number calculated by the correlation proposed by Gunn42 and Wen and Fan43 are 1.96 and 2.03, respectively, while the uneven distribution Peclet number was found in this study ranging from 0.25 to 2.51 (Figure 5). 8367



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Figure 12. Sequences of contours of gas-phase temperature (Tg) at different time (where, for example, 3.86eþ02 represents 3.86 102): Yin,ads = 2.25 103, Tg,in = 365[365 exp(0.352t0.666 1.654)] K, P = 1.09 atm, and F = 33.5 L min1.

In Figure 8, no significant variation of interstitial velocity at z = 0.10 m was observed when the r/R increased from 0 to 0.8, which was due to the relative uniform porosity near the column center. The inhomogeneity of porosity near the column wall (r/R ranges from 0.8 to 0.99) resulted in significant variation of velocity ranging from 0.85 to 1.1 m s1. This phenomenon has been confirmed by many investigators.44,45 The profile of concentration in the column was somewhat similar, but not identical, to that of velocity; this indicates that the velocity is a factor influencing the distribution of concentration but not the unique one. Regeneration of Methylene Chloride. The regeneration of methylene chloride by using nitrogen as purge gas after its adsorption in the active carbon packed column was simulated with the model, as shown in the previous section. The simulated regeneration curve was compared with the reported experimental data.22 (1) Concentration Profiles along the Column. The comparison between the simulated result and the experimental measurement (Figure 9) showed that the ratio of the outlet to the preadsorption inlet mole fraction of methylene chloride Yout/Yin,ads increased rapidly at the initial stage, reached a maximum value (4.1) at 16 min, then decreased to 1.0 at 35 min, and followed by continue to drop to zero. This roll-up phenomenon can be explained by the equilibrium effect caused by temperature changes. The serial contours of the mole fraction profile along the packed column are given in Figure 10, which shows the details of the behaviors of the mass transfer in the packed column. (2) Temperature Curve for Methylene Chloride Regeneration. The simulated radial temperature of the purge gas was averaged at different column heights (z = 0, 0.1, and 0.2 m, respectivly) during the regeneration process and was compared with experimental measured temperature curves as shown in Figure 11. As seen from the figure, each temperature curve consists of an ascending part and a relatively steady part. At the beginning instant of regeneration, the inlet temperature of the purge gas is 299 K, and then it is gradually increased to 399 K according to the expression given by Hwang. Following the progress of regeneration, the faster heat supply by incoming hot purge gas is greater than the heat needed for desorption and environmental loss; therefore, the gas temperatures at different column heights are raised sharply forming the ascending part of the temperature curve. When such condition is reached that most of the adsorbate have been desorbed and only a smaller part of the sensible heat of the purge gas is sufficient to balance the remaining heat of desorption and the heat loss, so that the gas temperature is maintained almost constant, forming the relatively

steady part of the temperature curve. In Figure 11, some deviations could be seen in the region of the ascending part profiles for the z = 0.1 and 0.2 m curves; Hwang explained that it might be due to the uncertainty in calculation of the heat of adsorption. Perhaps further explanation is the heat of desorption, which is assumed to be equal to the heat of adsorption in magnitude but opposite in sign, in the ascending part is overestimated, with the result that the measured temperatures of purge gas are higher than the simulated temperatures at z = 0.1 m before 10 min and z = 0.2 m before 20 min. After those times, the regeneration approaching the end and the heat needed for desorption gradually drop to zero; thus, the simulated temperatures are closely checked by the measurements. The serial contours of the temperature profile along the packed column are given in Figure 12, which demonstrates the details of the axial and radial temperature distributions of purge gas along the whole column.

’ CONCLUSIONS In this study, a rigorous method for simulating gas adsorption and regeneration processes with c2εc formulation for the closure of mass-transfer differential equation is proposed. With this model the simulation of the concentration distribution in an adsorption and regeneration column can be undertaking without assuming in advance the turbulent mass-transfer diffusivity. The simulated results for methylene chloride adsorption and regeneration agreed satisfactorily with the experimental measurement as reported from the literature. The wall effect was demonstrated by the concentration and velocity profiles. The mass-transfer diffusivity predicted by the present model shows deceasing tendency near the wall, but it turns to an increase in a small region adjacent the wall. The predicted turbulent Schmidt number and Peclet number distributions displayed unevenly throughout the column. Thus, the conventional way of assuming a constant Schmidt is not applicable for rigorous simulation. The simulated concentration contours at different time may be helpful to deepen the understanding of the process dynamics. The present model is especially useful for cases where the empirical means for finding the turbulent mass-transfer diffusivity is not available. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Tel.: þ86-22-27404732. Fax: 86-22-27404496. 8368



’ ACKNOWLEDGMENT We acknowledge financial support by the National Natural Science Foundation of China (Contract No. 20736005) and assistance by the staff in the State Key Laboratories of Chemical Engineering (Tianjin University). ’ NOMENCLATURE ap = surface area per unit volume of packed column, m1 aγ = constant depending on the ratio of the particle size to column size b = Langmuir isotherm constant, atm1 bh = heat convection parameters C = mean concentration (mass fraction) Cin = inlet concentration (mass fraction) cp,g, cp,s = specific heats of the gas phase and the solid phase, respectively, J kg1 K1 Cμ, C1ε, and C1ε = turbulence model constants for the velocity field Cc0, Cc1, Cc2, and Cc3 = turbulence model constants for the concentration field Ct0, Ct1, Ct2, and Ct3 = turbulence model constants for the temperature field c2 = concentration variance D = molecular diffusivity, m2 s1 Da = axial dispersion, m2 s1 Dt = turbulent mass diffusivity, m2 s1 Dp = pore dispersion, m2 s1 Dt,a = turbulent mass diffusivity in axial direction, m2 s1 Dt,r = turbulent mass diffusivity in radial direction, m2 s1 dH = hydraulic diameter of packing, m dp = nominal packing diameter, m dcol = inner diameter of the column, m Er = exponential decaying function F = flow rate, L min1 g = gravity acceleration, m s2 Gr = Grashof number (Gr = F2gβ(Tg T0)dcol3/μ2) h = heat-transfer coefficient from gas phase to packing, W m2 K1 hw = heat-transfer coefficient from gas phase to ambient, W m2 K1 hw1 = heat-transfer coefficient from gas phase to column wall, W m2 K1 hw2 = heat-transfer coefficient from column to ambient, W m2 K1 ΔH = heat of adsorption of adsorbate, J mol1 k = turbulent kinetic energy, m2 s2 kf = external mass-transfer coefficient, m s1 KG = overall mass-transfer coefficient, m s1 kg = thermal conductivity of gas, W m1 K1 kp = internal mass-transfer coefficient, m s1 ks = thermal conductivity of adsorbent particle, W m1 K1 n = heat convection parameters M = molecular weight of adsorbate, kg mol1 Nu = Nusselt number (Nu = hRp/kg) P = total pressure of gas phase in the column, atm Pet, Pea = turbulent and axial peclet numbers based on particle diameter (Pet = |u|dp/Dt, Pea = |u|dp/Da) Pr = Prandtl number (Pr = cp,gμ/kg) q, q0, q*, qs = adsorbate, initial adsorbate, equilibrium, and saturation adsorbate concentrations in solid phase, respectively, mol kg1 Qw1 = heat flux from gas phase to the inner wall, W m2 Qw2 = heat flux from outer wall to the envinorment, W m2 Qw = heat flux through column wall, W m2

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r = radial distance from the axis of the column, m R = inner radius of the column, m Rτ = velocity to concentration time scale ratio Rp = packing radius, m Rep, Recol = Reynolds numbers based on packing and column diameters, respectively (Rep = F|u|dp/μ, Recol = F|u| dcol/μ) Sc, Sct = Schmidt numbers based on molecular and turbulent diffusivities, respectively (Sc = μ/FD, Sct = μ/FDt) Sc = source of interphase mass transfer, kg m3 s1 Sm = source of interphase momentum transfer, N m3 STg = thermal source term of the gas phase, J m3 s1 STs = thermal source term of the solid phase, J m3 s1 t = time, min Tg = gas-phase temperature, K Tg,in = inlet temperature of the gas phase, K Ts = solid-phase temperature, K Tw1, Tw2 = temperatures of the inner and outer walls, respectively, K T0 = ambient temperature and initial temperature of the solid phase, K t2 = temperature variance u = gas interstitial velocity vector, m s1 ui, uj = gas interstitial velocities in axial and radial directions, m s1 u0 = gas fluctuating velocity, m s1 xi, xj = coordinates in axial and radial directions Y, Y* = mole fraction and equilibrium mole fraction of adsorbate in the gas phase, respectively Yin = inlet mole fraction of adsorbate in the gas phase for the adsorption step Yin,ads = preadsorption step inlet mole fraction for the desorption step yw = distance from the column wall, m z = height of packing measured from the gas-phase inlet of the column, m Z = total packing height of the column, m Greek Symbols

r, rt, reff = molecular, turbulent, and effective thermal diffusivities, respectively, m2 s1 β = volume expansibility of the air in the ambient, K1 ε = turbulent dissipation, m2 s3 εc = turbulent dissipation of the concentration, s1 εt = turbulent dissipation rate of temperature fluctuation, s1 γ = local column porosity γ¥ = porosity in an unbounded packing γp = particle porosity δij = Kronecker delta F = gas density, kg m3 Fg = total gas concentration, mol m3 Fs = apparent density of the adsorbent pellet, kg m3 μ, μt, μeff = gas molecular, turbulent, and effective viscosities, respectively, kg m1 s1 σk, σε, σc, σεc , σt, σεt = turbulence model constants for diffusion of k, ε, c2, εc, t2, εt

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